Answer: Step 1:Write the linear system of equations in matrix form as $AX=B;$

$\Rightarrow $ $\begin{bmatrix}1 & 2 & 6 \\ 3 & 4 & 1 \\ 6 & -1 & -1 \end{bmatrix} \begin{bmatrix}x \\ y \\ z \end{bmatrix}=\begin{bmatrix}22 \\ 26 \\ 19 \end{bmatrix}$

Step 2:Write Augmented matrix:

$\left[ \begin{array}{ccc|r} 1 & 2 & 6 & 22 \\ 3 & 4 & 1& 26 \\ 6 & -1 & -1& 19 \\ \end{array} \right] $ $\Rightarrow$$\left[ \begin{array}{ccc|r} 1 & 2 & 6 & 22 \ 0 & -2 &-17 & -40 \ 0 & -13 & -37& -113 \ \end{array} \right]$ $R_2 \rightarrow R_2-3R_1,R_3 \rightarrow R_3-6R_1$

$\Rightarrow $$\left[ \begin{array}{ccc|r} 1 & 2 & 6 & 22 \ 0 & 2 &17 & 40 \ 0 & 13 & 37& 113 \ \end{array} \right]$ $(-1)R_3,(-1)R_2$

$\Rightarrow$$\left[ \begin{array}{ccc|r} 1 & 2 & 6 & 22 \ 0 & 2 &17 & 40 \ 0 & 0 & -147/2& -147 \ \end{array} \right]$ $R_3 \rightarrow R_3-13/2(R_2)$

$\Rightarrow$$\left[ \begin{array}{ccc|r} 1 & 2 & 6 & 22 \ 0 & 2 &17 & 40 \ 0 & 0 & 1& 2 \ \end{array} \right]$ $R_3 \times(-2/147)$

$\Rightarrow$ $\left[ \begin{array}{ccc|r} 1 & 0 & -11 & -18 \\ 0 & 2 &17 & 40 \\ 0 & 0 & 1& 2 \\ \end{array} \right]$ $R_1 \rightarrow R_1-R_2$

$\Rightarrow$ $\left[ \begin{array}{ccc|r} 1 & 0 & 0 & 4 \\ 0 & 1 &0 & 3 \\ 0 & 0 & 1& 2 \\ \end{array} \right]$ $R_1 \rightarrow R_1+11R_3, R_2 \rightarrow R_2-17R_3, R_2 \rightarrow R_2/2$

$\Rightarrow$$\begin{bmatrix}1 & 0 & 0\ 0 & 1 & 0 \ 0 & 0 & 1 \end{bmatrix} \begin{bmatrix}x \ y \ z \end{bmatrix}=\begin{bmatrix}4 \ 3 \ 2 \end{bmatrix}$$\Rightarrow x=4, y=3, z=2.$   Answer.